Prediction method for the truck's fault time in open-pit mines based on exponential smoothing neural network

The transport truck is one of the important equipment for open-pit mines, and predicting the truck's fault time is of great significance in improving the economic benefits of open-pit mines. In this paper, we discuss the reason for the large prediction error of the exponential smoothing method. Then, we propose a novel nonlinear exponential smoothing method (ESNN) for predicting the truck's fault time, and demonstrate the equivalence between our approach and the neural network structure. Finally, based on the augmented Lagrange function, the solving method of ESNN is proposed. We conduct experiments on real-world datasets and our results demonstrate the effectiveness of ESNN in comparison to existing state-of-the-art methods. Our approach makes it easier for maintenance personnel to predict fault situations in advance and provides a basis for enterprises to develop preventive maintenance plans.


Method Exponential smoothing
Exponential Smoothing (ES) was formally proposed by Brown in 1959, who believed that the time status is stable so that the time series can be reasonably extended.Based on the above assumption, a weighted linear combination of observation data is used to calculate future predictions.The weight decreases exponentially with the further increase of past observations.The smallest weight is associated with the oldest observations.
According to the number of smoothing times, exponential smoothing methods include single exponential smoothing, second exponential smoothing, and third exponential smoothing.Single exponential smoothing is suitable for predicting data without clear trends or seasonal patterns.The second exponential smoothing is suitable for predicting the future value of the data with a linear trend.The third exponential smoothing is appropriate for the data with an obvious seasonal pattern.Due to the lack of clear trends and seasonal patterns in the observed data during equipment faults, single exponential smoothing is chosen as the basis for the algorithm.The future prediction is defined as: where ⌢ y t+1 is the predicted value at the time t + 1 , y t is the observed value, ⌢ y t is the predicted value at the time t , α is the smoothing parameter.
The smoothing parameter is often selected based on experience.However, few works are applied to predict the fault time of equipment.Therefore, there is no empirical value for reference.Besides, as time goes by, the distribution of random variables may change.If α is a fixed value, it will also impact the prediction accuracy.

Neural network model based on exponential smoothing (ESNN)
The classic exponential smoothing method takes the weighted average of the observed data as the future prediction.Therefore, when the observed values fluctuate within a small range of the average value, the exponential smoothing method presents a small error.However, if the variance of the observed values is large, the single exponential smoothing method may cause significant errors at certain time points.
The analysis shows that the main reason for the higher local error is that the weighted average of the observations is directly used as the prediction result.The problem can be solved by establishing a mapping between the weighted average of the observations and the future prediction.We also use the weighted average value as the independent variable to fit the future prediction.
Assume that there is a mapping between the weighted average and the future prediction of the observations such that xt+1 = f (x t+1 ) , where x t+1 =αx t + (1 − α)x t .We build a neural network model equivalent to the above assumptions to establish the mapping and the parameter, as shown in Fig. 1.
From Fig. 1, we can see that: Combining Eq. (2), we can see that: (1) Then, this neural network model is equivalent to the single exponential smoothing model.However, if the activation function g is applied to the node xt+1 .Equation ( 5) is the previously defined mapping xt+1 = f (x t+1 ) .Training ESNN can solve the parameters in Eq. (5).Assuming the activation function is a hyperbolic tangent function in the node xt+1 .The hyperbolic sine function has monotonicity and periodicity, which is an activation function.At the same time, it has high complexity and the ability to introduce nonlinearity into neural networks, making it suitable for this paper.The mean square error of ESNN on the training set where, Besides, the weights w 11 , w 12 , w 21 , w 22 , v 11 , v 12 , v 21 , v 22 , s need to satisfy the first equation in Eq. ( 4).Transform the training of the ESNN model into an optimization process for solving constraint equations.
In Eq. ( 14), the parameters w, v, s and θ are updated in each iteration.The updating formula for parameters and δ is defined as: where ρ(ρ > 1) is the step size.The details of the solution of the ESNN model are described in Algorithm 1.

Test data and environment
The 8-year (2010-2018) maintenance data of the TR100 (Shenhua Baorixile Energy Co., Ltd.) is adopted as the test data.The test data is grouped according to the truck number, and the fault interval time is computed (the next maintenance time minus the previous maintenance time) to obtain the sequence of time between faults.The data has been tested for stationarity, and the results indicate that the test data is a non-stationary sequence.
The ESNN algorithm is implemented in C# language and executed in Visual Studio 2013.All experiments are performed on the same workstation (CPU: E5-2620, memory: 32 GB).

Settings
For the time series of each truck's fault interval, ESNN and Autoregressive Integrated Moving Average (ARIMA) are used to predict the time between truck faults under four different sliding window sizes (win size).Among them, ARIMA belongs to the classic model in time series analysis, which is widely used due to its high prediction efficiency, small error, and other advantages, and it also conforms to the prediction scenario.Repeat the experiment ten times under each sized sliding window and average the obtained results to produce the final output.In addition, five performance indicators are applied in the experiment: absolute error (AE), relative error(RE), Akaike's information criterion(AIC), corrected Akaike's information criterion(AICc), and Schwarz's Bayesian information criterion(BIC).These five information errors are divided into two traditional metrics (AE and RE) and information criterion metrics (AIC, AICc, BIC).Traditional metrics measure the predictive performance of the model, while the information criteria are used to evaluate the fitting effect of the model.
The testing for stationarity shows that the test data is the non-stationary series.Therefore, the differential order of ARIMA is set as 0. The parameters of ESNN are described in Table 2.
In the experiment, we chose sliding window sizes of 6, 7, 8, and 9.They meet the size of the data in this paper and improve prediction accuracy.

Evaluation
Figure 2 shows the AE and RE of four different sliding window sizes.In a summary, it can be observed that the AE of ESNN is smaller than ARIMA, which proves that our approach can improve prediction accuracy.When win size = 6, the AE of ARIMA is slightly higher than ESNN, which is evident in the interval [0, 200].However, the AE of ARIMA is slightly lower than ESNN; When win size = 7, the AE of ESNN slightly lower than ARIMA in the interval [0, 50], but ESNN is slightly higher than ARIMA in the interval [250, 280], and the average levels of the AE of ESNN and ARIMA are similar in general; When win size = 8, the AE of ESNN is slightly higher than ARIMA, and it is obvious in the interval [0,60], [80, 120], and [160,180].The above phenomenon can be further observed in Table 3. Besides, it can be observed that the AE of ESNN is the lowest when win size = 8.The main reason for the above phenomenon is that the number of data points that ESNN needs to fit increases with the size of the sliding window.Each data point affects the connection weight and neuron threshold of the ESNN in the current sliding window.If the fault interval time of future time points is only related to the fault interval time of the nearest time point, then early time points in the same sliding window will hurt the update of connection weights and neuron thresholds.
Table 3 summarizes the results on four types of sliding window sizes for all algorithms.It can be seen that the difference in AE between ESNN and ARIMA is only 0.16, and the RE of ESNN is equal to ARIMA.This indicates that the AE of ESNN is similar to the RE of ARIMA, making it difficult to distinguish which algorithm is better.Therefore, we choose AIC, AICc, and BIC for further analysis.Table 3 shows that the AIC and AICc of ESSN are lower than ARIMA when win size = 8 and win size = 9.The BIC of ESSN is lower than ARIMA when win size = 6.In addition, the average values of AIC, AICc, and BIC in ESNN are the lowest.The gain of ESNN (according to the reduction in AIC, AICc, and BIC) concerning ARIMA is 2, 23, and 5. Overall, ESNN outperforms ARIMA in most indicators.The results show that ESNN has relatively stable prediction error and higher accuracy than ARIMA.This further proves the superiority of ESNN in predicting truck failure time in open-pit mines.

Conclusion
(1) We analyze the principle of the exponential smoothing model and the problems in predicting the truck's fault time, and design a neural network model based on single exponential smoothing method.Compared with existing state-of-the-art methods, ESNN achieves higher accuracy and provides an effective approach for predicting the truck's fault time in open-pit mines.Although the ESNN algorithm has better performance compared to other algorithms, it still has some limitations.For example, the ESNN results are not very prominent for some experimental sliding windows.Due to the structural characteristics of nonlinear exponential smoothing algorithms, the optimized neural network may have a curse of dimensionality in complexity, which makes ESNN unsuitable for predicting long-term fault problems.Therefore, we will conduct further research on these issues in the future, as follows: (1) In the future, we will improve the nonlinear exponential smoothing algorithm to better adapt to the neural network model.(2) In the future, we will reduce the complexity of neural networks to improve prediction accuracy and applicability to higher latitude data.

Figure 2 .
Figure 2. The absolute error and relative error under different moving windows.

Table 2 .
The parameters of ESNN.

Table 3 .
The statistic results of different moving windows.